A New Algorithm for the Inverse of Periodic k Banded and Periodic Anti k banded Matrices

TL;DR

This paper introduces a new LU-based algorithm for efficiently computing inverses, determinants, and solving linear systems involving periodic k banded and anti k banded matrices, useful in differential equations.

Contribution

The paper presents a novel LU factorization-based algorithm specifically designed for periodic k banded and anti k banded matrices, including inverse, determinant, and linear system solutions.

Findings

Algorithm successfully computes inverses of the matrices.

Determinants of these matrices are derived.

Linear systems with these matrices are effectively solved.

Abstract

In this study, an algorithm for computing the inverse of periodic k banded matrices, which are needed for solving the differential equations by using the finite differences, the solution of partial differential equations and the solution of boundary value problems is obtained and the inverses of periodic anti k banded matrices are computed. In addition, the determinant of these type of matrices and the solution of linear systems having these coefficient matrices are investigated. When obtaining this algorithm, the LU factorization is used.The algorithm is implementable to the CAS (Computer Algebra Systems) such as Maple and Mathematica.